DESIGN OF SPIRAL INDUCTORS USING EVOLUTIONARY OPTIMISATION L. TOUGIANNIDIS1, S. IEZEKIEL1, R.D. POLLARD1 AND G.D. HALIKIAS2 1 UNIVERSITY OF LEEDS, UK 2 CITY UNIVERSITY, LONDON, UK KEYWORDS: Spiral Inductors, Layout, Optimisation, Synthesis, Integrated Inductors ABSTRACT: Simplex and evolutionary optimisation methods are used to determine the optimum topology and geometric parameters of inductors, in order to satisfy electrical design requirements (quality factor, resonance frequency and inductance). The technique simplifies considerably the design of integrated spiral inductors on silicon. Various topologies are considered using an accurate, lumped-element equivalent model, which is sufficiently simple to allow simulations of several hundred iterations to be performed rapidly. The algorithm is highly successful in determining the optimal design parameters, by avoiding convergence to local minima and without requiring good initial estimates. The accuracy of the method is validated both against measured data and results from electromagnetic simulations.
INTRODUCTION Spiral inductors are critical components in many circuit blocks used in wireless applications and RF transceivers. The recent advances in CMOS technology have renewed the interest in the design of spiral inductors since their performance has improved to an acceptable level for many non-critical designs. Low-cost consumer electronics and telecommunication products have been increasingly using integrated passives but they have yet to be improved in order to satisfy performance-critical design blocks. For those applications, more exotic variants of CMOS technology or even GaAs processes are required. However, the research society has put a lot of effort in order to further improve their performance and analyse their behaviour. This effort has been concentrated into two mainly aspects. The first is the optimisation of the layout of integrated spiral inductors [1] that can be done using electromagnetic simulation software and mathematical formulae without the need of any alterations on the available fabrication process. The second part of the effort is concentrated in improving the fabrication process or adjusting and tweaking it to suit the design of spiral inductors. In the later case, the goal usually is to minimise the parasitic effects of the lossy substrate by either maximising its resistivity [2], inserting metal [3] or polymer [4] shields or even removing the substrate underneath the inductor [5]. Much of that effort however, has been skewed by criticism from the industry due to the fact that some or most of these solutions involve increased cost or additional fabrication processes. Although many electromagnetic CAD programs can accurately simulate integrated spiral inductors using a variety of algorithms, the design of an inductor is complex (fig. 1) because its performance not only depends on the metal strips and its layout (as for a lumped inductor), but also on the substrate material properties and the influence of adjacent elements. The overall complexity of the loss mechanism can often lead to conflicting objectives when choosing design parameters. Moreover, it is
impossible to predict a-priori the geometry (topology and layout) of a spiral inductor that will satisfy the electrical specifications. Typically, one must consider several different inductor structures and perform numerous electromagnetic simulations using a trialand-error method on the geometry of the spiral in order to find a solution. A single electromagnetic simulation using ADS Momentum® often takes a few hours. Thus, the design of a spiral inductor and the optimisation of its parameters via trial-and-error requires a few days, and there is no guarantee of an acceptable (let alone optimum) solution. There is, therefore, a need for a tool to help one choose an appropriate inductor topology and optimise its geometric parameters.
Fig. 1. Losses occurring on an integrated inductor In this work, a novel design methodology based on genetic algorithms (GADM) is presented, that creates and optimizes the layout of integrated spiral inductors in order to meet electrical specifications. Genetic algorithms are very successful in many optimisation problems, especially when the cost function to be minimised exhibits multiple local minima, for which traditional “descent-direction” algorithms often fail.
SYNTHESIS OF SPIRAL INDUCTORS The first stage of the method involves the specification of electrical parameters (Q and inductance), together with information on material properties and layout limitations (fig. 2). The cost function is defined as the sum of the absolute
Fig.2. The input / output parameters of the GADM percentage error in quality factor and inductance, relative to the specified values. The genetic algorithm then generates the first set of proposed layouts (number of turns, width, spacing, thickness of the segments and internal radius of the spiral) and assesses them using the cost function. This is a measure of the fitness of the population of the current geometries and signifies the degree that they satisfy the design requirements. The verification process involves frequency-domain simulation of the lumped-element model (fig 3) corresponding to the proposed layout.
Fig.3. Lumped-element equivalent model This model is fully described in the literature [6], with the additional assumption that substrate resistivity exceeds 0.1 µ.cm. The model omits SiO2 capacitance accounts for Si capacitance and skin effects, while being compatible with differential and symmetric inductors. This is being used as a starting point guess for the optimiser since it is a very simple and fast to simulate model. However, as the optimiser reaches to the required solution, this model is being swapped with more accurate and application specific models that are chosen on the criteria of frequency-of-use, substrate characteristics, required shape etc. This helps to provide an even more accurate picture of how the geometry should be in order to meet the electrical requirements. Figure 4, 5 and 6 demonstrate the validity of these lumpedelement models against measured data from a series of fabricated inductors.
Fig. 4. Measured and model simulated data (2 turns)
Fig. 5. Measured and model simulated data (3 turns)
Fig. 6. Measured and model simulated data (4 turns) Next, the genetic algorithm selects the best designs based on their performance. It applies genetic operations on them (fig 7) in order to improve their “fitness” and forwards them to the next stage of the optimisation algorithm. Some of these operations are namely crossover where half of the genetic code (or real geometrical diamensions in the solution domain) of the candidate’s DNA is being exchanged with one half from a different candidate. Mutation is another function that allows one part of the DNA of a selected portion of the generation to be chanced on a random basis with irrelevant information. Inversion is the complete alteration of the binary code of the DNA by interchanging all the digits from 1 to 0 and visa-versa. Finally, selection is another random function that ensures that there would always be new DNA information in the population that could affect its fitness in a positive way. This process is described in figure 7.
Fig. 7. Overview of the design process Every population of candidate solutions to the problem, after is being modified with the above operations is being converted back to real dimensions and numbers and forms the fitting parameters for the lumped-element model. Therefore, it represents new geometries that would later be simulated in the frequency domain and yield the electrical performance.
Each iteration uses a population of 30 to 50 different design configurations and usually takes about 50 to 100 iterations to converge, giving a computation time of about 15 minutes. The fitness of the designs improves rapidly until a small error is reached (Fig 8), at which point the electrical parameters are very close (user-defined) to the specified values. The algorithm terminates when the value of the cost function falls below a specified tolerance level.
meet almost any feasible electrical parameter specification. The GADM software allows for the efficient search of an inductor layout that meets design specifications in terms of electrical parameters such as L and Q, and satisfies geometric (area) constraints. Thus, the design time is dramatically reduced, typically from hours when electromagnetic simulations are used, down to only a few minutes.
Fig. 8. The fitness improves as the error reduces
RESULTS The cost function plot (fig. 9) shows that the error surface is highly complex even in the simple case of only two varying physical parameters (spacing, width); here only the error on the inductance optimisation is considered. Traditional descent-based optimisation methods would likely fail, especially in the case when six parameter values vary (fig. 2) and the error is a function of the peak quality factor, inductance and resonance. The program also generates Y- and Z-parameters, inductance, resistance (fig. 11) and quality factor (fig. 12) versus frequency. Results obtain for 12 different structures indicate close agreement between measured and simulated data. (S22 and S21 plots are omitted due to reciprocity.
Fig. 10. The success rate of the synthesizer
Fig. 11. The fittest candidates of the 25th population
Fig. 9 The error as function of width & spacing only The algorithm converges quickly to the optimum layout and matches all the feasible design specifications, with an error less than 4%. Figure 10 demonstrates the success and failure rate for a number of attempted syntheses. The program will either meet both the quality and inductance criteria by 4% or by 20% tolerance. In these series of tests, there were attempts that do not have solutions with any geometry and therefore by largest, the failed runs were not a failure of the code. The designs were verified using ADS Momentum. The whole process requires no more than 15 minutes and produces layouts that can
Fig. 12. The fittest candidates of the 25th population
Acknowledgements The authors would like to thank Dr L. Albasha, C. Clifton and P. Shadwell from Sony Semiconductor Europe for their continuous support.
THE AUTHORS L. Tougiannidis, Dr S. Iezekiel and Prof. R.D. Pollard are with the Institute of Microwaves and Photonics, University of Leeds, Leeds, UK
[email protected] Dr G.D. Halikias is with the Department of Electrical, Electronic and Informations Engineering, City University, London, UK
[email protected]
REFERENCES [1] L. Tougiannidis, S. Iezekiel, R.D. Pollard and G.D. Halikias, “An Efficient Two-layer Spiral Inductor Configuration” IEEE, International Conference Mixed Design of Integrated Circuits and Systems, 2005
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